Remark 3.6.3.2. Let $f: X \rightarrow Y$ be a continuous function between topological spaces. Then $f$ is a weak homotopy equivalence of topological spaces if and only if $\operatorname{Sing}_{\bullet }(f)$ is a weak homotopy equivalence of simplicial sets. This is a special case of Proposition 3.1.6.13, since the simplicial sets $\operatorname{Sing}_{\bullet }(X)$ and $\operatorname{Sing}_{\bullet }(Y)$ are Kan complexes (Proposition 1.2.5.8).
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